
where \(M_{\rm P}\equiv 1/\sqrt{8\pi G}=2.44\times 10^{18}\)  GeV is the reduced Planck mass, \(R_{\rm GB}^2\equiv R^2-4R_{\mu \nu }R^{\mu \nu }+R^{\mu \nu \rho \sigma }R_{\mu \nu \rho \sigma }\)  is the Gauss-Bonnet term (the 4-dimensional Euler density).
The coupling function is chosen to beThis form of the coupling is motivated in microscopic physics as follows.
In calculable examples of type II and heterotic compactifications, the \(R_{\rm GB}^2\)  coupling to moduli typically occurs as 1-loop gravitational threshold corrections, which are determined by the spectrum of BPS states [1]}, [2]}.
If one of the moduli \(\varphi \)  traverses a wall separating two domains with different spectra, the coupling function \(\xi (\varphi )\)  behaves as a step function across the wall.
If the domain wall has a finite thickness, \(\xi (\varphi )\)  is modeled by a smeared step function (REF ).
\(\xi (\varphi )&=\xi _0\tanh \left[\xi _1(\varphi -\varphi _c)\right]\,,\) 
